The following external R packages are used:

library(dplyr)
library(gganimate) # 'gifski' and 'png' package have to be installed
library(tidyr)
include_graphics <- knitr::include_graphics

1. Introduction

This is to illustrate the “jumping frog” Markov chain 1 using gganimate package.

2. A jumping frog

Say there is a frog jumping across 20 lily pads numbered from 1 to 20 which are arranged clockwise in a circle. Suppose \(X_n\) is the pad number at the \(n\)th jump. Let’s define the “jumping frog” Markov chain \(\{ X_n \}_{n = 0}^{\infty}\) as a Markov chain whose state space is \(S = \{ 1, 2, \dots, 20\}\), with the initial state \(X_0 = 20\) and initial probabilities \(v_{20} = P(X_0 = 20) = 1\) and \(v_s = 0 \text{ }\) for all \(s = 1, \dots, 19\), and trainsition probabilities \(p_{ij} = P(X_n = j \text{ | } X_{n - 1} = i)= \frac{1}{3}\) whenever \(j = i - 1, i\), or \(i + 1\).

Long story short, a Java Applet regarding this frogwalk written by Prof. Rosenthal at the University of Toronto doesn’t work anymore. This is the effort to simulate this Markov chain using R and gganimate package.

With initial parameters below, we can simulate this Markov chain and populate X with some generated \(X_n\)’s:

set.seed(1024)

# Settings
max_iter <- 1000 # The maximum n
number_of_pads <- 20
initial_state <- 20
p_ij = rep(1, 3) / 3 # Prob of moving left, staying, and moving right

# Populate X
state_space <- 1:number_of_pads
cmf_p_ij = cumsum(p_ij)
X <- NULL
X[1] <- state_space[initial_state] # initial state
for (i in 2:(max_iter + 1)) {
    choice <- runif(1) # choice ~ Unif(0, 1)
    if (0 <= choice && choice < cmf_p_ij[1]) { # move left
        if (X[i - 1] == number_of_pads) {
            X[i] <- 1
        } else {
            X[i] <- X[i - 1] + 1 
        }
    } else if (cmf_p_ij[1] <= choice && choice < cmf_p_ij[2]) { # stay
        X[i] <- X[i - 1]
    } else { # move right
        if (X[i - 1] == 1) {
            X[i] <- number_of_pads
        } else {
            X[i] <- X[i - 1] - 1
        }
    }
}

state_df will be used to create a tibble that indicates the pad number the frog is located at each iteration. state_space will be used as x-axis, pads_point as y-axis (which is just a dummy), and iterations as a facetting variable:

state_df <- tibble(
  iterations = rep(
    0:max_iter, 
    each = number_of_pads
  ),
  state_space = rep(state_space, max_iter + 1),
  pads_point = rep(1, number_of_pads * (max_iter + 1))
)

comparison_df is a tibble that acutally indicates where the frog is at at each iteration. state_space and iterations will be variables that state_df and comparison_df have in common, and value will be an indicator:

comparison_df <- tibble(state_space)
for (i in 1:(max_iter + 1)) {
  mutating_code <- paste0(
    "comparison_df <- comparison_df %>% ",
    "mutate(`", i - 1, "` = state_space == X[i])"
  )
  eval(parse(text = mutating_code))
}
comparison_df <- comparison_df %>% 
    gather(iterations, value, -state_space) %>%
    mutate(iterations = as.integer(iterations))

We shall now left_join state_df and comparison_df using state_space and iterations columns, and create the whole states_df that is ready for generating a .gif file:

states_df <- state_df %>%
    left_join(comparison_df, by = c("state_space", "iterations"))

In our gganimate usage, we are basically creating a fast slide show of all the facets. iterations is our facetting variable, which is used in transition_states. Whenever the frog is located at the pad, the image will denote such a state with a red dot, and the rest of the pads will be coloured as black.

## Warning: Both x_mod and y_mod are 0. No position change will occur.

## Warning: Both x_mod and y_mod are 0. No position change will occur.
anim <- ggplot(
    states_df, 
    aes(
        x = factor(state_space), y = pads_point, 
        color = value
    )
) + 
    geom_point() +
    scale_color_manual(values = c("black", "red")) +
    coord_polar(start = .15) +
    labs(title = "Iteration: {frame_time}", x = "", y = "") +
    theme(legend.position = "none") +
    transition_time(iterations) +
    enter_drift() + 
    exit_drift()
states_gif <- animate(
  anim, fps = 20,
  duration = 78,
  start_pause = 2,
  end_pause = 2
)
anim_save('frog.gif', states_gif)

Here’s the .gif file:

include_graphics('frog.gif')

Here’s a slower version:

states_gif2 <- animate(
  anim, fps = 20,
  duration = 226,
  start_pause = 2,
  end_pause = 2
)
anim_save('frog_slower.gif', states_gif2)
include_graphics('frog_slower.gif')

If your sole purpose is to display, then saving the image file and using include_graphics() is unnecessary. For that purpose, you can just use animate().

Session info

R session info:

sessionInfo()
## R version 3.6.0 (2019-04-26)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 17763)
## 
## Matrix products: default
## 
## locale:
## [1] LC_COLLATE=English_Canada.1252  LC_CTYPE=English_Canada.1252   
## [3] LC_MONETARY=English_Canada.1252 LC_NUMERIC=C                   
## [5] LC_TIME=English_Canada.1252    
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
## [1] tidyr_0.8.3     gganimate_1.0.3 ggplot2_3.1.1   dplyr_0.8.1    
## 
## loaded via a namespace (and not attached):
##  [1] Rcpp_1.0.1        pillar_1.4.0      compiler_3.6.0   
##  [4] plyr_1.8.4        prettyunits_1.0.2 prettydoc_0.2.1  
##  [7] tools_3.6.0       progress_1.2.2    digest_0.6.18    
## [10] evaluate_0.13     tibble_2.1.1      gtable_0.3.0     
## [13] pkgconfig_2.0.2   rlang_0.3.4       yaml_2.2.0       
## [16] xfun_0.7          withr_2.1.2       stringr_1.4.0    
## [19] knitr_1.23        hms_0.4.2         grid_3.6.0       
## [22] tidyselect_0.2.5  glue_1.3.1        R6_2.4.0         
## [25] rmarkdown_1.12    purrr_0.3.2       tweenr_1.0.1     
## [28] farver_1.1.0      magrittr_1.5      scales_1.0.0     
## [31] htmltools_0.3.6   assertthat_0.2.1  colorspace_1.4-1 
## [34] stringi_1.4.3     lazyeval_0.2.2    munsell_0.5.0    
## [37] crayon_1.3.4

  1. Rosenthal, J. (2019, April 05). STA447/2006 (Stochastic Processes) Lecture Notes, Winter 2019. Retrieved May 21, 2019, from http://probability.ca/jeff/teaching/1819/sta447/notes.pdf↩